EQUATION
, in Algebra, an expression of equality between two different quantities; or two quantities, whether simple or compound, with the mark of equality between them: as ; &c. When the things, or two sides of the equation, are the same, the expression becomes an Identity, as 5 = 5, or a = a, &c. Sometimes the quantities are placed all on one side, and made equal to 0 or nothing on the other side; as : which is no more than setting down the difference of two equal quantities equal to nothing.
The character or sign usually employed to denote an Equation, is =, which is placed between the two equal quantities, called the two sides of the Equation.
The Terms of an Equation, are the several quantities or parts of which it is composed. Thus, of the Equation , the terms are a, b, and c: and the tenor or import of the expression is, that some quantity represented by c, is equal to two others represented by a and b.
Equations are either Simple or Affected.
A Simple Equation is that which has only one power of the unknown quantity: as , &c; where x denotes the unknown quantity, and the other letters known ones. But
An Assected, or Adsected Equation contains two or more different powers: as , &c. |
Again, Equations are denominated from the highest power contained in them; as quadratic, cubic, biquadratic, &c. Thus,
A Quadratic Equation, is that in which the unknown quantity rises to two dimensions, or to the square or 2d power: as .
A Cubic Equation, is that in which the unknown quantity is of three dimensions, or rises to the cube or 3d power: as .
A Biquadratic Equation, is that in which the unknown quantity is of 4 dimensions, or rises to the 4th or biquadratic power: as .
And so for other higher orders of Equations.
The Root of an Equation, is the value of the unknown letter or quantity contained in it. And this value being substituted in the terms of the Equation instead of that letter or quantity, will cause both sides to vanish, or will make the one side exactly equal to the other. So the root of the Equation , is 10; because that using 10 for x, it becomes .
Every Equation has as many roots as it has dimensions, or as it contains units in the index of the highest power, when the powers are all reduced to integral exponents. So the simple Equation of the 1st power, has only one root; but the quadratic has 2, the cubic 3, the biquadratic 4, &c. Thus the two roots of this equation are 1 and 3; for either of these substituted for x makes x2-4x come out equal to - 3. Also the three roots of , are 2, 5, and -3; as will appear by substituting each of these instead of x in the equation, which will make all the terms on one side equal to the other side. And so of others.
The Relation between the Roots of Equations, and the Coefficients of their Terms.——In every Equation, when the terms are ranged in order according to the order of the powers, the greater before the less; the first term or highest power freed from its coefficient, by dividing all the terms by it, and all brought to one side, and made equal to nothing on the other side, when it will appear in this form, ; then the relations between the roots and coefficients, are as follow:
1st, The coefficient a of the 2d term, is equal to the sum of all the roots.
2d, The coefficient b of the 3d term, is equal to the sum of all the products of the roots that can be made by multiplying every two of them together. In like manner,
3d, The coefficients c, d, e, &c, of the following terms, are respectively equal to the sum of the products of the roots made by multiplying every three together, or every four together, or every five together, &c, the signs of all the roots being changed. All which will appear below, in the Generation of Equations.
The Roots of Equations are Positive or Negative, and Real or Imaginary. Thus, the two roots of the Equation , are 1 and 3, real and both positive; but the roots of the Equation , are 2, 5, & -3, are real, two positive and one negative; and the roots of the equation , are 1 and - 1/2 ± 1/2 √(-39), one real and two imaginary.
The Generation of Equations, is the multiplying of certain assumed simple equations together, to produce compound ones, with intent to shew the nature of these; a method invented by Harriot, which is this: Suppose x to denote the unknown quantity of any equation, and let the roots of that equation, or the values of x, be, a, b, c, d, &c; that is x = a, and x = b, and x = c, &c; or , &c; then multiply these last equations together, thus,
Now the roots of these equations are a, b, c, d, &c; and it is obvious that the sum of all the roots is the coefficient of the 2d term, the sum of all the products of every two is the coefficient of the 3d term, the sum of all the products of every three that of the 4th term, and so on, to the last term, which is the continual product of all the roots.
Reduction of Equations, is the transforming or changing them to their simplest and most commodious form, to prepare them for finding or extracting their roots. The most convenient form is, that the terms be ranged according to the powers of the unknown letter, the highest power foremost next the left hand, and that term to have only + 1 for its coefficient; also all the terms containing the unknown letter to be on one side of the equation, and the absolute known term only on the other side.
Now this reduction chiefly respects the first term, or that which contains the highest power of the unknown quantity; and the general rule for reducing it is, to consider in what manner it is involved or connected with other quantities, and then perform the counter or op- | posite relation or operation; for every operation is undone or counteracted by the reverse of it; as addition by subtraction, multiplication by division, involution by evolution, &c: then bring all the unknown terms to one side, and the known term to the other side, changing the signs, from + to -, or from - to +, of those terms which are changed from one side to the other; and lastly divide by the coefficient of the first term, with its sign.
This is finding the value or values of the unknown letter in an Equation, the rules for which are various, according to the degree of the Equation.
Having reduced the equation as above, by bringing the unknown terms to one side, and the known ones to the other, freeing the former from radicals and fractions, by their counter operations, and lastly dividing by the coefficients of the unknown quantity, the value of it is then found: as in the first and 2d examples of reduction above given.
These are usually found by what is called completing the square; which consists in squaring half the coefficient of the 2d term, and adding it to both sides of the equation; for then the unknown side is a complete square of a binomial, and the other side consists only of known quantitles. Therefore, extract the root on both sides, so shall the root of the first side be a binomial, one part of which is the unknown letter, and the other a known or given quantity, and the root of the other side is taken either + or -, since the square of either of these is the same given quantity: lastly, bringing over the known part of the binomial root to the other side, with a contrary sign, gives the two roots or values of the unknown letter sought.
Thus, if be a general quadratic Equation, 2a being the coefficient of the 2d term, and b2 the absolute known term, both with their signs. Then, a is half that coefficient, and a2 its square; which being added, gives ; and the root extracted gives ; then, transposing a, it is , the two roots, or values of x.
A Cubic Equation is that in which the unknown letter ascends to the 3d power; as .
The 2d term of every Cubic Equation being taken away, those equations may all be reduced to this form, ; and the general value of one root is . This rule is usually called Cardan's, because first published by him, but it was invented both by Scipio Ferreus, and Nich. Tartalea, by whom it was communicated to Cardan. See the article Algebra.
When the 2d term is negative, or the equation of this form, , the radical √((1/4)b2 + (1/27)a3) becomes √((1/4)b2-(1/27)a3), which will be imaginary or impossible when (1/27)a3 is greater than (1/4)b2, for √((1/4)b2-(1/27)a3) will then be the square root of a negative quantity, which is impossible: and yet, in this case, the root x is a real quantity; though algebraists have never been able to sind a real finite general expression for it. And this is called the Irreducible or Impracticable Case.
This case may indeed be resolved by the trisection of an arc or angle; or by any of the usual methods of converging; or by general expressions in infinite series. See Saunderson's Algebra, pa. 713; Philos. Trans. vol. 18, pa. 136, or Abr. vol. 1, pa. 87; also vol. 70, pa. 415. See also the article Cubic Equations.
Mr. Cotes observes, in his Logometria, pa. 29, that the solution of all cubic Equations depends either upon the trisection of a ratio, or of an angle. See this method explained in Saunderson's Alg. p. 718. |
Biquadratic Equations, or those that are of 4 dimensions, are resolved after various methods. The first rule was given by Lewis Ferrari, the companion of Cardan, which is one of the best. A 2d method was given by Des Cartes, and another by Mr. Simpson and Dr. Waring. For the explanation of which, see BIQUADRATIC Equations.
There is no general rule to express algebraically the roots of Equations above those of the 4th degree; and therefore methods of approximation are here made use of, which, though not accurately, are yet practically true. Some of these excel in ease and simplicity, and others in quickness of converging. Among these may be reckoned first, Double Position, or Trial-and-Error, both in respect of ease and universality, as it applies in the simplest manner to all sorts of Equations whatever, not excepting even exponential ones, radical expressions of ever so complex a form, expressions of logarithms, of arches by the sines or tangents, of arcs of curves by the abscisses, or any other fluents, or roots of fluxional Equations. For an explanation of this and other methods of converging to the roots of equations, by Halley, Newton, Raphson, &c, &c, see APPROXINATION, and Converging.
Besides the methods above adverted to, there have been some others, given in the Memoirs of several Academies, and elsewhere. As, by M. Daniel Bernoulli, in the Acta Petropolitana, tom. 3, p. 92; and by M. L. Euler, in the same, vol. 6, New Series, and tom. 5, p. 63 & 82; by Mr. Thos. Simpson, in his Essays, p. 82; in his Dissertations, p. 102; in his Algebra, p. 158; and in his Select Exercises, p. 215.
Absolute Equation. See Absolute.
Adfected, or Affected Equation. See Affected.
Differential Equation, is the Equation of Differences or Fluxions.
Eminential Equation. See Eminential.
Exponential Equation, one in which the exponents of the powers are variable or unknown quantities. See Exponential.
Fluential Equation, is the Equation of the fluents.
Fluxional Equation, is the Equation of the fluxions.
Literal Equation, is a general Equation expressed in letters, as contradistinguished from a
Numeral Equation, one expressed in numbers.
Transcendental Equation. See Transcendental.
Equation, in Astronomy, as Annual Equation, is either of the mean motion of the sun and moon, or of the moon's apogee and nodes.
The Annual Equation of the sun's centre being given, the other three corresponding annual Equations, will be also given, and therefore a table of the first will serve for all of them. Thus, if the annual Equation of the sun's centre, taken from such a table for any time, be called s; and if (1/10)s = A, and (1/6)s= B; then shall the other annual Equations for that time be thus,
, that of the moon's mean motion; and , that of the moon's apogee; and , that of her nodes.
And here note, that when s, or the Equation of the sun's centre, is additive; then m is negative, a is positive, and n is negative. But on the contrary, when s is negative or subductive; then m is positive, a negative, and n positive.
There is also an Equation of the moon's mean motion, depending on the situation of her apogee in respect of the sun; which is greatest when the moon's apogee is in an octant with the sun, and is nothing at all when it is in the quadratures of syzygies. This Equation when greatest, and the sun in perigee, is 3′ 56″. But it is never above 3′ 34″ when the sun is in apogee. At other distances of the sun from the earth, this Equation when greatest, is reciprocally as the cube of that distance. But when the moon's apogee is any where out of the octants, this Equation grows less, and is mostly, at the same distance between the earth and sun, as the sine of double the distance of the moon's apogee from the next quadrature or syzygy, is to radius. This is to be added to the moon's motion while her apogee passes from a quadrature with the sun to a syzygy; but is to be subtracted from it, while the apogee moves from the syzygy to the quadrature.
There is moreover another Equation of the moon's motion, which depends on the aspect of the nodes of the moon's orbit with respect to the sun: and this is greatest when her nodes are in octants to the sun, and quite vanishes when they come to their quadratures or syzygies. This Equation is proportional to the sine of double the distance of the node from the next syzygy or quadrature; and at the greatest is only 47″. This must be added to the moon's mean motion while the nodes are passing from the syzygies with the sun to their quadratures; but subtracted while they pass from the quadratures to the syzygies.
From the sun's true place subtract the equated mean motion of the lunar apogee, as was shewn above, the remainder will be the annual argument of the said apogee; from whence the eccentricity of the moon and the 2d Equation of her apogee may be compared. See Theory of the Moon's motions, &c.
Equation of the Centre, called also Prosthapheresis, and Total Prosthapheresis, is the difference between the true and mean place of a planet, or the angle made by the lines of the true and mean place; or, which amounts to the same, between the mean and equated anomaly.
The greatest Equation of the Centre may be obtained by finding the sun's longitude at the times when he is near his mean distances, for then the difference will give the true motion for that interval of time: next find the sun's mean motion for the same interval of time; then half the difference between the true and mean motions will shew the greatest Equation of the Centre.
For Example, by observations made at the Royal Observatory at Greenwich, it appears that at the following mean times the sun's longitudes were as annexed; viz,
| Mean times. | Sun's longitudes. | |||
| 1769 Oct. 1 at 23h 49m 12s | 6s | 9° | 32′ | 0.6″ |
| 1770 Mar. 29 at 0 4 50 | 0 | 8 | 50 | 27.5 |
| dif. of time 178d 0 15 38; true dif. lon. | 5 | 29 | 18 | 27 |
| Therefore | 175° | 27′ | 21″ | of mean motion, |
| answers to | 179 | 18 | 27 | of true motion; |
| their difference is | 3 | 51 | 6 | |
| and its half | 1 | 55 | 33 |
To find the Equation of the Centre, or to resolve Kepler's problem, is a very troublesome operation, especially in the more eccentric orbits. How this is to be done, has been shewn by Newton, Gregory, Keil, Machin, La Caille, and others, by methods little differing from one another; which consist chiefly in finding a certain intermediate angle, called the eccentric anomaly; having known the mean anamoly, and the dimensions of the sun's orbit. The mean anomaly is easily found, by determining the exact time when the sun is in the aphelion, and using the following proportion, viz, As the time of a tropical revolution, or solar year, Is to the interval between the aphelion and given time, So is 360 degrees, to the degrees of the mean anomaly. Or it may be found by taking the sun's mean motion at the given time out of tables.
To find the Eccentric Anomaly, say, As the aphelion distance, Is to the perihelion distance; So is the tangent of half the mean anomaly, To the tangent of an arc.
Which arc added to half the mean anomaly, gives the eccentric anomaly. Then,
To find the True Anomaly, say, As the square root of the aphelion distance, Is to the square root of the perihelion distance; So is the tangent of half the eccentric anomaly, To the tangent of half the true anomaly.
Then, the difference between the true and mean anomaly, gives the Equation of the Centre, sought. Which is subtractive, from the aphelion to the perihelion, or in the first 6 signs of anomaly; and additive, from the perihelion to the aphelion, or in the last 6 signs of anomaly; and hence called Prosthapheresis.
By this problem a table may easily be formed. When the Equations of the Centre for every degree of the first 6 signs of mean anomaly are found, they will serve also for the degrees of the last 6 signs, because equal anomalies are at equal distances on both sides of their apses. Then set these equations orderly to their signs and degrees of anomaly; the first 6 being reckoned srom the top of the table downwards, and signed subtract; the last 6, for which the same Equations serve, in a contrary order, being reckoned from the bottom upwards, and marked add. Let also the difference between every adjacent two Equations, called Tabular Differences, be set in another column. Hence, from these Equations of the Centre, augmented or diminished by the proportional parts of their respective tabular differences, for any given minutes and seconds, may easily be deduced Equations of the Centre to any mean anomaly proposed. Robertson's Elem. of Navig. book 5, p. 286, 290, 295, and 308, where such a table of Equations is given.
The late excellent Mr. Euler has particularly considered this subject, in the Mem. de l'Acad. de Berlin, tom. 2, p. 225 & seq. where he resolves the following problems:
1. To find the true and mean anomaly corresponding to the planet's mean distance from the sun; that is, where the planet is in the extremity of the conjugate axis of its orbit.
2. The eccentricity of a planet being given, to find the eccentric anomaly corresponding to the greatest Equation.
3. The eccentricity being given, to find the mean anomaly corresponding to the greatest Equation.
4. From the same data, to find the true anomaly corresponding to this Equation.
5. From the same data, to find the greatest Equation.
6. The greatest Equation being given, to find the eccentricity.
Mr. Euler observes, that this problem is very difficult, and that it can only be resolved by approximation and tentatively, in the manner he mentions: but if the eccentricity be not great, it may then be found directly from the greatest Equation. Thus, if the greatest Equation be = m, and the eccentricity = n; then is Whence by reversion Where the greatest equation m must be expressed in parts of the radius, which may be done by reducing the angle m into seconds, and adding 4.6855749 to the log. of the resulting number, which will be the log. of the number m.
The mean anomaly to which this greatest equation corresponds, will be Whence, if 90° be added to 5/8 of the greatest Equation, the sum will be the mean anomaly sufficiently exact.
Mr. Euler subjoins a table, by which may be found the greatest Equations, with the mean and eccentric anomalies corresponding to these greatest Equations for every 100th part of unity, which he supposes equal to the greatest eccentricity, or when the transverse and distance of the foci become infinite. The last column of the table gives also the logarithm of that distance of the planet from the sun where its Equation is greatest. By means of this table, any eccentricity being given, by interpolation will be found the corresponding greatest Equation. But the chief use of the table is to determine the eccentricity when the greatest Equation is known; and without this help Mr. Euler thinks the problem cannot be resolved.
Equation of Time, denotes the difference between mean and apparent time, or the reduction of the apparent unequal time, or motion of the sun or a planet, to equal and mean time, or motion; or the Equation of time is the difference between the sun's mean motion, and his right ascension. Apparent time is that which takes its beginning from the passage of the sun's centre over the meridian of any place; and had the sun no | motion in the ecliptic, or was his motion reduced to the equator or in right ascension uniform, he would always return to the meridian after equal intervals of time. But his apparent motion in the ecliptic being continually varying, and his motion in right ascenfion being rendered farther unequal on account of the obliquity of the ecliptic to the equator, from these causes it arises that the intervals of his return to the meridian become unequal, and the sun will gradually come too slow or too soon to the meridian for an equable motion, such as that of clocks and watches ought to be; and this retardation or acceleration of the sun's coming to the meridian, is called the equation of time.
Now, computing the celestial motions according to equal time, it is necessary to turn that time back again into apparent time, that they may correspond to observation: on the contrary, any phenomenon being observed, the apparent time of it must be converted into equal time, to have it correspond with the times marked in the astronomical tables.
The Equation of time is nothing at four different times in the year, when the whole mean and unequal motions exactly agree; viz, about the 15th of April, the 15th of June, the 31st of August, and the 24th of December: at all other times the sun is either too fast or too slow for mean, equal, or clock time, by a certain number of minutes and seconds, which at the greatest is 16′ 14″, and happens about the 1st of November; every other day throughout the year having a certain quantity of this difference belonging to it; which however is not exactly the same every year, but only every 4th year; for which reason it is necessary to have 4 tables of this Equation, viz, one for each of the four years in the period of leap years. Instead of these, may be here inserted, as follows, one general equation of time, according to the place of the sun, in every point of the ecliptic: where it is to be observed, that the sign of the ecliptic is placed at the tops of the columns, and the particular degree of the sun's place, in each sign, in the first and last columns; and in the angle of meeting in all the other columns, is the equation of time, in minutes and seconds, when the sun has any particular longitude: supposing the obliquity of the ecliptic 23° 28′, and the sun's apogee in 9° of .
| A Table of the Equation of Time, for every Degree of the Sun's Longitude. | |||||||||||||||||||||||||
| Deg. | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | Deg. | ||||||||||||
| m | s | m | s | m | s | m | s | m | s | m | s | m | s | m | s | m | s | m | s | m | s | m | s | ||
| 0 | 7 | +36 | 1 | -9 | 3 | -51 | 1 | +13 | 5 | +57 | 2 | +20 | 7 | -38 | 15 | -31 | 13 | -33 | 1 | -11 | 11 | +28 | 14 | +19 | 0 |
| 1 | 7 | 17 | 1 | 23 | 3 | 47 | 1 | 26 | 5 | 59 | 2 | 4 | 7 | 58 | 15 | 39 | 13 | 17 | 0 | 42 | 11 | 45 | 14 | 13 | 1 |
| 2 | 6 | 58 | 1 | 36 | 3 | 41 | 1 | 40 | 6 | 0 | 1 | 48 | 8 | 19 | 15 | 46 | 13 | 0 | 0 | 12 | 12 | 1 | 14 | 6 | 2 |
| 3 | 6 | 39 | 1 | 48 | 3 | 37 | 1 | 53 | 6 | 1 | 1 | 31 | 8 | 40 | 15 | 52 | 12 | 42 | 0 | +17 | 12 | 17 | 13 | 59 | 3 |
| 4 | 6 | 20 | 2 | 0 | 3 | 32 | 2 | 7 | 6 | 1 | 1 | 14 | 9 | 1 | 15 | 57 | 12 | 23 | 0 | 46 | 12 | 32 | 13 | 51 | 4 |
| 5 | 6 | 1 | 2 | 11 | 3 | 26 | 2 | 20 | 6 | 0 | 0 | 56 | 9 | 21 | 16 | 2 | 12 | 4 | 1 | 16 | 12 | 46 | 13 | 43 | 5 |
| 6 | 5 | 42 | 2 | 22 | 3 | 19 | 2 | 33 | 5 | 59 | 0 | 38 | 9 | 41 | 16 | 6 | 11 | 44 | 1 | 45 | 12 | 59 | 13 | 34 | 6 |
| 7 | 5 | 24 | 2 | 32 | 3 | 12 | 2 | 45 | 5 | 57 | 0 | 20 | 10 | 1 | 16 | 9 | 11 | 23 | 2 | 14 | 13 | 12 | 13 | 24 | 7 |
| 8 | 5 | 5 | 2 | 42 | 3 | 4 | 2 | 58 | 5 | 54 | 0 | 1 | 10 | 20 | 16 | 11 | 11 | 1 | 2 | 43 | 13 | 24 | 13 | 14 | 8 |
| 9 | 4 | 47 | 2 | 51 | 2 | 56 | 3 | 11 | 5 | 51 | 0 | -18 | 10 | 39 | 16 | 13 | 10 | 39 | 3 | 11 | 13 | 35 | 13 | 3 | 9 |
| 10 | 4 | 28 | 3 | 0 | 2 | 47 | 3 | 23 | 5 | 47 | 0 | 37 | 10 | 57 | 16 | 13 | 10 | 16 | 3 | 39 | 13 | 45 | 12 | 51 | 10 |
| 11 | 4 | 9 | 3 | 8 | 2 | 38 | 3 | 35 | 5 | 42 | 0 | 57 | 11 | 15 | 16 | 13 | 9 | 53 | 4 | 7 | 13 | 54 | 12 | 39 | 11 |
| 12 | 3 | 50 | 3 | 16 | 2 | 29 | 3 | 46 | 5 | 37 | 1 | 17 | 11 | 33 | 16 | 12 | 9 | 29 | 4 | 35 | 14 | 2 | 12 | 27 | 12 |
| 13 | 3 | 32 | 3 | 23 | 2 | 19 | 3 | 58 | 5 | 31 | 1 | 38 | 11 | 5<*> | 16 | 10 | 9 | 5 | 5 | 2 | 14 | 9 | 12 | 14 | 13 |
| 14 | 3 | 13 | 3 | 30 | 2 | 8 | 4 | 9 | 5 | 24 | 1 | 58 | 12 | 8 | 16 | 7 | 8 | 40 | 5 | 29 | 14 | 16 | 12 | 0 | 14 |
| 15 | 2 | 55 | 3 | 36 | 1 | 57 | 4 | 19 | 5 | 17 | 2 | 19 | 12 | 25 | 16 | 4 | 8 | 14 | 5 | 56 | 14 | 22 | 11 | 46 | 15 |
| 16 | 2 | 37 | 3 | 41 | 1 | 46 | 4 | 29 | 5 | 9 | 2 | 40 | 12 | 41 | 16 | 0 | 7 | 48 | 6 | 22 | 14 | 27 | 11 | 31 | 16 |
| 17 | 2 | 19 | 3 | 46 | 1 | 35 | 4 | 39 | 5 | 1 | 3 | 1 | 12 | 57 | 15 | 55 | 7 | 22 | 6 | 48 | 14 | 31 | 11 | 16 | 17 |
| 18 | 2 | 1 | 3 | 50 | 1 | 23 | 4 | 48 | 4 | 52 | 3 | 22 | 13 | 12 | 15 | 49 | 6 | 55 | 7 | 13 | 14 | 35 | 11 | 1 | 18 |
| 19 | 1 | 43 | 3 | 53 | 1 | 11 | 4 | 57 | 4 | 43 | 3 | 44 | 13 | 27 | 15 | 42 | 6 | 28 | 7 | 37 | 14 | 38 | 10 | 46 | 19 |
| 20 | 1 | 26 | 3 | 56 | 0 | 59 | 5 | 5 | 4 | 33 | 4 | 5 | 13 | 42 | 15 | 35 | 6 | 0 | 8 | 1 | 14 | 40 | 10 | 30 | 20 |
| 21 | 1 | 9 | 3 | 58 | 0 | 46 | 5 | 13 | 4 | 22 | 4 | 26 | 13 | 56 | 15 | 26 | 5 | 32 | 8 | 24 | 14 | 41 | 10 | 14 | 21 |
| 22 | 0 | 52 | 4 | 0 | 0 | 34 | 5 | 20 | 4 | 11 | 4 | 47 | 14 | 9 | 15 | 17 | 5 | 4 | 8 | 47 | 14 | 42 | 9 | 58 | 22 |
| 23 | 0 | 36 | 4 | 1 | 0 | 21 | 5 | 27 | 3 | 59 | 5 | 9 | 14 | 21 | 15 | 7 | 4 | 36 | 9 | 9 | 14 | 41 | 9 | 41 | 23 |
| 24 | 0 | 20 | 4 | 1 | 0 | 8 | 5 | 33 | 3 | 46 | 5 | 30 | 14 | 33 | 14 | 56 | 4 | 8 | 9 | 31 | 14 | 40 | 9 | 24 | 24 |
| 25 | 0 | 4 | 4 | 1 | 0 | +5 | 5 | 39 | 3 | 33 | 5 | 52 | 14 | 44 | 14 | 44 | 3 | 39 | 9 | 53 | 14 | 39 | 9 | 6 | 25 |
| 26 | 0 | -11 | 4 | 0 | 0 | 19 | 5 | 44 | 3 | 19 | 6 | 13 | 14 | 53 | 14 | 31 | 3 | 10 | 10 | 14 | 14 | 37 | 8 | 48 | 26 |
| 27 | 0 | 26 | 3 | 59 | 0 | 31 | 5 | 48 | 3 | 4 | 6 | 35 | 15 | 5 | 14 | 17 | 2 | 41 | 10 | 34 | 14 | 34 | 8 | 30 | 27 |
| 28 | 0 | 40 | 3 | 57 | 0 | 46 | 5 | 52 | 2 | 50 | 6 | 56 | 15 | 14 | 14 | 3 | 2 | 11 | 10 | 53 | 14 | 30 | 8 | 12 | 28 |
| 29 | 0 | 53 | 3 | 54 | 0 | 59 | 5 | 55 | 2 | 35 | 7 | 17 | 15 | 23 | 13 | 48 | 1 | 41 | 11 | 11 | 14 | 25 | 7 | 54 | 29 |
| 30 | 1 | 9 | 3 | 51 | 1 | 13 | 5 | 57 | 2 | 20 | 7 | 28 | 15 | 31 | 13 | 33 | 1 | 11 | 11 | 28 | 14 | 19 | 7 | 36 | 30 |
The Equations with +, are to be added to the apparent time, to have the mean time, those with -, are to be subtracted from apparent time, to give the mean time.
The preceding mark, whether + or -, at the top of any column, belongs to all the numbers or equations in that column till the sign changes; after which, the remainder of the column belongs to the contrary sign.
The Equation answering to any point of longitude between one degree and another, or any number of minutes or parts of a degree, is to be found by proportion in the usual way, viz, as 1° or 60′, to that number of minutes, so is the whole difference in the Equation from the given whole degree of longitude to the next degree, to the proportional part of it answering to the given number of minutes.
See Tables of the Equation of Time computed for every year, in the Nautical Almanac, by a method proposed and illustrated by Dr. Maskelyne, the astronomer royal, viz, by taking the difference between the sun's true right ascension and his mean longitude, corrected by the Equation of the equinoxes in right ascension, and turning it into time at the rate of 1 minute of time to 15′ of right ascension. Philos. Trans. vol. 54, p. 336.
Equation of a Curve, is an Equation shewing the nature of a curve by expressing the relation between any absciss and its corresponding ordinate, or else the relation of their fluxions, &c. Thus, the Equation to the circle, is , where a is its diameter, x any absciss, or part of that diameter, and y the ordinate at that point of the diameter; the meaning being, that whatever abseiss is denoted by x, then the square of its corresponding ordinate will be ax-x2. In like manner the Equation of the ellipse is , of the hyperbola is , of the parabola is px = y2. Where a is an axis, and p the parameter.
And in like manner for any other curves.
This method of expressing the nature of curves by algebraical equations, was first introduced by Des Cartes, who, by thus connecting together the two sciences of algebra and geometry, made them mutually assisting to each other, and so laid the foundation of the greatest improvements that have been made in every branch of them since that time. See Des Cartes's Geometry; also Newton's Lines of the 3d Order, and many other similar works on curve lines, by several authors.
